2464 CHAPTER 73. THE HARD ITO FORMULA, IMPLICIT CASE

The most significant feature of the last term is that it is a local martingale. The term⟨BZ,Z⟩L2

will be discussed later, as will the meaning of the stochastic integral.The idea is that

(Z ◦ J−1

)∗BX ◦ J has values in L2(Q1/2U,R

)and so it makes sense

to consider this stochastic integral. To see this, BX ∈W ′ and

(Z ◦ J−1)∗ ∈L2

(W ′,

(JQ1/2U

)′)and so (

Z ◦ J−1)∗BX ∈(

JQ1/2U)′

and so (Z ◦ J−1)∗BX ◦ J ∈L2

(Q1/2U,R

)=(

Q1/2U)′

Note that in general H ′ = L2 (H,R) because if you have {ei} an orthonormal basis in H,then for f ∈ H ′,

∑i

∣∣(R−1 f ,ei)∣∣2 = ∑

i|⟨ f ,ei⟩|2 = ∥ f∥2

H ′ .

The main item of interest relative to this stochastic integral will be a statement about itsquadratic variation. It appears to depend on J but this is not the case because the otherterms in the formula do not.

73.3 Preliminary ResultsHere are discussed some preliminary results which will be needed. From the integral equa-tion, if φ ∈ Lq (Ω;V ) and ψ ∈C∞

c (0,T ) for q = max(p,2) ,∫Ω

∫ T

0

((BX)(t)−B

∫ t

0Z (s)dW (s)−BX0

)ψ′φdtdP

=∫

Ω

∫ T

0

∫ t

0Y (s)ψ

′ (t)dsφdtdP

Then the term on the right equals∫Ω

∫ T

0

∫ T

sY (s)ψ

′ (t)dtdsφ (ω)dP =∫

Ω

(−∫ T

0Y (s)ψ (s)ds

)φ (ω)dP

It follows that, since φ is arbitrary,∫ T

0

((BX)(t)−B

∫ t

0Z (s)dW (s)−BX0

)ψ′ (t)dt =−

∫ T

0Y (s)ψ (s)ds

in Lq′ (Ω;V ′) and so the weak time derivative of

t→ (BX)(t)−B∫ t

0Z (s)dW (s)−BX0